Router-level community structure of the Internet Autonomous Systems

The Internet is composed of routing devices connected between them and organized into independent administrative entities: the Autonomous Systems. The existence of different types of Autonomous Systems (like large connectivity providers, Internet Service Providers or universities) together with geographical and economical constraints, turns the Internet into a complex modular and hierarchical network. This organization is reflected in many properties of the Internet topology, like its high degree of clustering and its robustness. In this work we study the modular structure of the Internet router-level graph in order to assess to what extent the Autonomous Systems satisfy some of the known notions of community structure. We observe that most of the classical community detection methods fail to detect the Autonomous Systems as communities, mainly because the modular structure of the Internet (as that of many complex networks) is much richer than what can be captured by optimizing a global functional: Autonomous Systems have largely variable sizes, structures and functions. Classical methods are severely affected by resolution limits and by the heterogeneity of the communities; even when using multiresolution methods, there is no single resolution at which most of the communities can be captured. However, we show that multiresolution methods do find the community structure of the Autonomous Systems, but each of them has to be observed at the correct resolution level. Then we develop a low-complexity multiresolution modularity optimization algorithm that finds communities at different resolution levels in a continuous scale, in one single run. Using this method, we show that with a scarce knowledge of the node affiliations, multiresolution methods can be adjusted to retrieve the Autonomous Systems, significantly improving the results of classical single-resolution methods. Finally, in the light of our results, we discuss recent work concerning the use of a priori information to find community structure in complex networks.